Geometric Challenge Problem 1620: Perpendicular Chords and Opposite Areas, a Hidden Linear Invariant

Share your proof or solution in the comments below.
Target Audience: K-12, Honors Geometry, and College Mathematics Education.

Uncover the mathematical elegance hidden within a circle. In Problem 1620, we explore how two perpendicular chords partition a circle into four curvilinear regions to reveal a remarkable hidden area difference invariant.
Explore the full theorem and illustrated diagrams by clicking the image below.

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Proposed Solution
We invite students, teachers, and math enthusiasts to share their insights. This challenge involves perpendicular chords, Archimedes' Theorem, and curvilinear areas that can be unlocked using geometry.

How to contribute:
Post your step-by-step proof in the comments below. Feel free to:

• Describe the theorems applied (e.g., Intersecting Chords, Archimedes).
• Share a link to your dynamic construction (GeoGebra, Desmos).

Be the first to submit a formal solution for this area invariant problem!

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Please use the box below to Enter your Comment or Solution. You can use plain text or provide links to your digital proofs.

Geometry Hidden in the Chino Copper Mine

From space, the El Chino Copper Mine in New Mexico reveals a surprising world of circles, arcs, symmetry, and geometric patterns.

Viewed through Google satellite imagery, massive circular settling tanks, curved roads, and industrial structures transform this mining landscape into a striking example of real-world geometry.

Click below to reveal the complete details.

Explore more at GoGeometry.

Geometric Challenge Geometry Problem 1617: The Concyclic Hexagon and the Power of Intersecting Circumcircles

Share your proof or solution in the comments below.
Target Audience: K-12, Honors Geometry, and College Mathematics Education.

Discover the hidden symmetry of intersecting systems in Problem 1617. This challenge moves away from simple tangency to explore a more complex configuration where two circles overlap, creating a remarkable "Concyclic Hexagon."
Explore the full theorem and interactive diagrams by clicking the illustration below.

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Proposed Solution
We invite students, teachers, and enthusiasts to share their proofs. This classic challenge can be approached using synthetic geometry.

How to contribute:
Post your step-by-step proof in the comments below. Feel free to:

• Describe your construction and properties applied (centroid/medians).
• Provide a link to a diagram (GeoGebra, Desmos, etc.) if you have one.

Be the first to submit a formal solution!

Ready to contribute?
Please use the box below to Enter your Comment or Solution. You can use plain text or provide links to your diagrams.

Geometry Problem 1610: Area of a Curvilinear Triangle in a Rhombus

Challenging Geometry Problem 1610. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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Geometry Problem 1609: Radius of a Semicircle Tangent to Two Sides of a Triangle

Challenging Geometry Problem 1609. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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Geometry Problem 1608: Right Triangle Area from Incircle Tangent Segments and Hypotenuse Perpendiculars

Challenging Geometry Problem 1608. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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Geometry Problem 1607: Circle, Chords, Tangents, 60 Degree Angle

Challenging Geometry Problem 1607. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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Geometry Problem 1605: Triangle, Angle Bisector, and Circumcircle

Challenging Geometry Problem 1605. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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Geometry Problem 1604: Can You Find FG?

Challenging Geometry Problem 1604. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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Geometry Problem 1603: Can You Find the Hidden Distance AG?

Challenging Geometry Puzzle: Problem 1603. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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Geometry Problem 1602: Find the Distance from the Semicircle’s Center to the Incenter

Challenging Geometry Puzzle: Problem 1602. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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Geometry Problem 1597: Circle-Triangle Intersection Problem

Challenging Geometry Puzzle: Problem 1597. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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Geometry Problem 1588: Unlocking the Tangent: Prove the Fifth of the Radius in a Sector with Semicircle

Challenging Geometry Puzzle: Problem 1588. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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Geometric Problem1584: Proving a Ratio with the Incircle, Tangency Points, and Angle Bisector

Challenging Geometry Puzzle: Problem 1584. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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Problem-Solving Challenge 1583: Prove Lines CK and BL are Parallel

Challenging Geometry Puzzle: Problem 1583. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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Geometry Problem 1582: Prove That Angles AFD and AEF Are Equal in This Secant Problem.

Challenging Geometry Puzzle: Problem 1582. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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Geometry Problem 1579: Involving perpendicular diameters, tangents, and triangle areas.

Challenging Geometry Puzzle: Problem 1579. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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Geometry Problem 1574: Triangle with Three Circles through a Point and the Concyclicity

Challenging Geometry Puzzle: Problem 1574. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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Geometry Problem 1573: Find the Angle at K

Challenging Geometry Puzzle: Problem 1573. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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Geometry Problem 1568: Prove Concyclicity of Points B, D , H, J in a Triangle ABC

Challenging Geometry Puzzle: Problem 1568. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.

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